Neighbor distinguishing total choice number of sparse graphs via the Combinatorial Nullstellensatz

Let G = (V,E) be a graph and ϕ: V ∪E → {1, 2, · · ·, k} be a total-k-coloring of G. Let f(v)(S(v)) denote the sum(set) of the color of vertex v and the colors of the edges incident with v. The total coloring ϕ is called neighbor sum distinguishing if (f(u) ≠ f(v)) for each edge uv ∈ E(G). We say that ϕ is neighbor set distinguishing or adjacent vertex distinguishing if S(u) ≠ S(v) for each edge uv ∈ E(G). For both problems, we have conjectures that such colorings exist for any graph G if k ≥ Δ(G) + 3. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs, which is denoted by mad (G). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that these two conjectures hold for sparse graphs in their list versions. More precisely, we prove that every graph G with maximum degree Δ(G) and maximum average degree mad(G) has chΣ″(G) ≤ Δ(G) + 3 (where chΣ″(G) is the neighbor sum distinguishing total choice number of G) if there exists a pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k,m) \in \{ (6,4),(5,\tfrac{{18}} {5}),(4,\tfrac{{16}} {5})\}$$\end{document} such that Δ(G) ≥ k and mad (G) <m.